Published on Web 05/07/2002
Conformational Distribution Function of a Disaccharide in a Liquid Crystalline Phase Determined Using NMR Spectroscopy ¨ ,‡ Go ¨ ran Widmalm,‡ and Arnold Maliniak*,§ Baltzar Stevensson,§ Clas Landersjo DiVision of Physical Chemistry and Department of Organic Chemistry, Arrhenius Laboratory, Stockholm UniVersity, S-106 91 Stockholm, Sweden Received January 30, 2002
Proteins, nucleic acids, and carbohydrates all show conformational flexibility, but the extent is dependent on the structure and their environment. The motion in saccharides, in particular at the glycosidic linkage, defined by torsion angles φ and ψ is of importance for molecular properties and biological function. Thus, the approximation of a single molecular structure is certainly an oversimplified picture. To obtain complete information about the molecular structure we desire to determine the conformational distribution function, P(φ,ψ). Analysis of molecular conformations has for a considerable time relied on either the nuclear Overhauser effect (NOE), spin-spin (J) couplings, or a combination of these two. Recently, the application of dilute liquid crystalline phases (bicelles) as solvents enabled a slight net orientation of nonspherical “solute” molecules and therefore determination of through-space magnetic dipoledipole (DD) interactions. These provide a powerful tool for molecular structure analysis of saccharides in ordered phases.1-4 The DD interactions depend on the spin-spin distances and on the orientations of the internuclear vectors with respect to the external magnetic field. This means that the dipolar coupling is a valuable probe of long-range order and molecular structure. To extract useful information from the experimental DD couplings in a flexible molecule, we need a theoretical tool to be used in the analyses. Several such approaches have been considered for the interpretation of dipolar couplings. The simplest possible model assumes that only a small set of minimum-energy structures is populated. More realistic models allow for continuous bond rotations. Two approaches that have been frequently used for interpretations of dipolar couplings in bulk liquid crystals are: (i) the additive potential model (AP)5 and (ii) the maximum entropy method (ME).6 Both methods have been successfully applied in experimental and computational studies of the orientational order and molecular conformations in mesogenic molecules.7-9 The models suffer, however, from serious limitations: the AP method requires an a priori knowledge of the functional form of the torsional potential, relevant for the investigated molecular fragment. The ME approach though, gives the flattest possible distribution consistent with the experimental data set, which results in an incorrect description of systems with low orientational order.6,9 These two limitations have an obvious relevance for investigations of saccharides in dilute liquid crystals: we do not have the torsional potential function for the glycosidic linkage, and the orientational order is indeed very low. Here we present a novel approach for construction of the conformational distribution function P(φ,ψ) from the NMR parameters. The procedure, which is valid in the low-order limit, was * To whom correspondence should be addressed. E-mail: arnold.maliniak@ physc.su.se. § Division of Physical Chemistry. ‡ Department of Organic Chemistry.
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constructed as a combination of the AP and ME approaches, subsequently referred to as the APME method. In particular, the intraresidue dipolar couplings were used to determine the orientational order, while the conformational distribution function P(φ,ψ) was constructed from the interresidue DD- and J couplings, together with NOEs. We apply our analysis to R-L-Rhap-(1f2)-R-L-RhapOMe, shown in Figure 1, which is a model for part of the O-antigen repeating unit of the lipopolysaccharide from pathogenic Shigella flexneri bacteria. The saccharide exhibits motion over the glycosidic linkage as established from a preliminary analysis of intraring dipolar couplings using the generalized degree of order (GDO) approach.1,10 The GDOs in the two rigid rings differ by a factor of 1.2 (ϑR ) 0.0059 and ϑR′ ) 0.0072), whereas identical values are expected for rigidly connected fragments. The general expression for conformation dependent dipolar couplings, Dij(φ,ψ) can be written (in Hz) as:5
Dij(φ,ψ) )
µ0
γiγjp
∑cos θaij cos θbijSlab(φ,ψ)
16π2 r3ij(φ,ψ) a,b
(1)
where a,b ) (x,y,z) refers to an arbitrary coordinate frame fixed in one of the two rigid units, l ) R,R′, and θaij is the conformationdependent angle between the internuclear vector and the a-axis. Similarly, the conformation-dependent elements of the order matrix are denoted Slab(φ,ψ). The analysis of intraring couplings is in principle simple, because we do not need to explicitly consider the φ,ψ dependence. The interpretation of interresidue couplings, however, requires an expression for a molecular ordering matrix where both fragments contribute. To obtain Slab(φ,ψ), we consider the AP5 model where the singlet orientational distribution function (ODF), P(β,γ,φ,ψ), is related to the potential of mean torque. Note that the angles β and γ define molecular orientation in the director frame. This potential is determined by the expansion coefficients, lab, which depend on the orientational order and the segmental (R,R′) anisotropic interactions. In the limit of low-molecular order, the ODF can be Taylor-expanded and truncated after the second term. This results in the following expression for the order parameter:
SRab(φ,ψ) )
1
(Rab +
5RT
Taµ(φ,ψ)R′ ∑ µνTbν(φ,ψ)) µ,ν
(2)
where µ,ν ) (x,y,z) in the frame of R′, lab is defined in units of RT, and Taµ is an element of the φ,ψ-dependent transformation matrix that relates coordinate systems fixed in rings R and R′. An equivalent relationship is obtained for SR′ ab(φ,ψ) by interchanging the indices R and R′. Note that two important assumptions have 10.1021/ja025751a CCC: $22.00 © 2002 American Chemical Society
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Figure 1. Schematic of R-L-Rhap-(1f2)-R-L-Rhap-OMe. Comparison of observed and calculated NMR parameters for the disaccharide (DD- and J couplings in Hz; proton-proton distances rNOE in Å). Anisotropic medium: 5% DMPC/DHPC (q ) 3) in D2O at 37 °C. NMR experiments performed at 14.1 T: 2D 1JC,H modulated 13C,1H CT-HSQC,14 2D 1H,1H COSY,4,15 1D DPFGSE 1H,1H T-ROESY,16 and 1D Hadamard 3JC,H spectroscopy.17
been made:5 (i) each rigid segment (ring) makes its own contribution to the ordering potential, and (ii) lab parameters are conformationindependent. Thus, the conformation dependence of the ordering tensor is achieved by transformation to a common frame (Taµ(φ,ψ) in eq 2). In practice, the low-order approximation leads to an expression where the angles β and γ can be readily integrated out and the ODF becomes dependent on the φ,ψ angles only, that is, P(β,γ,φ,ψ) f P(φ,ψ). We are now in a position to construct a probability distribution function for the conformational variables φ and ψ. Of course, there are several options for the functional form of such a distribution. In the APME approach we decided on the distribution function which is derived from the maximum-entropy principle.6 The ME method provides the least biased and flattest distribution that is consistent with an experimental set of data. Thus, the conformational distribution function takes the form:
P(φ,ψ) )
1 Z
{
exp -
λkl ∑ij λijDij(φ,ψ) - ∑ 6 kl
1
rkl(φ,ψ)
-
λmn3Jmn(φ,ψ) ∑ mn
}
(3)
where Z is a normalization factor and the λ’s are adjustable parameters that can be determined by bringing calculated NMR parameters into agreement with those experimentally determined. Here λij, λkl, and λmn are related to the motionally averaged interresidue DD couplings, NOEs, and 3JC,H couplings, respectively. The recently developed Karplus-type relationship was used for interpretation of the trans-glycosidic 3JC,H couplings.11 Determination of a unique conformation-dependent order tensor requires at least five independent dipolar couplings in each ring. The intraresidue DD couplings were used to obtain the lab parameters by combining eqs 1-3 and using 〈Dintra ij 〉 ) ∫Dintra (φ,ψ)P(φ,ψ) dφ dψ. The interring DD couplings (five), ij together with the NOEs (four) and the 3JC,H couplings (two) were used to determine the λ parameters in eq 3. In the analysis 27 experimental points and 21 parameters (11 λ and 10 values) were employed in determination of P(φ,ψ). Note that the fitting procedure must be carried out simultaneously for all the parameters (λ and ). It is so, because of the conformation dependence of the ordering tensor used for calculation of all the DD couplings. Numerical fitting was performed using an in-house written computer code based on the MATLAB subroutine fminu. A complication that needs to be mentioned is the sign of the five interring H-H couplings. The J contribution to these spinspin interactions12 is essentially zero, and we were therefore not able to infer the sign of the dipolar interaction. Thus, in the analysis we considered 32 possible combinations. The fitting error for several
Figure 2. Contour map derived for the disaccharide, where φ ) H1′-C1′-O2-C2, and ψ ) C1′-O2-C2-H2. The distribution functions, P(φ,ψ), were determined using: (a) the APME approach (eq 3) and (b) 3J couplings and NOEs (only the two last terms of eq 3).
combinations was large, and these were therefore excluded from further considerations. In the final analysis we were able to identify a few sets of dipolar couplings that resulted in similar fitting errors. We performed a test of the APME approach by calculating the ordering tensors in each ring using the intraresidue DD couplings only, and by superimposing these tensors. This attempt to analyze all the experimental data using a single molecular conformation resulted in a much larger rmsd than for the APME procedure. The large error originates from poor agreement for the interring parameters. We conclude therefore, that a proper analysis of the molecular structure requires a distribution of conformations. In addition, we determined P(φ,ψ) using the 3JC,H couplings and NOEs, but excluding the DD couplings. The conformational distribution functions P(φ,ψ) are shown in Figure 2. The highly populated regions are in fact consistent with previously reported results derived from computer simulations of the same molecule.13 The distribution functions are very similar, indicating that the APME approach is consistent with the classical analysis of molecular conformations based on 3J couplings and NOEs. Figure 1 shows the correlation between measured and calculated NMR parameters using the APME distribution. The agreement is indeed very good. We believe that the APME method will be applicable as an important tool in structure determination of flexible molecules in dilute liquid crystals. Acknowledgment. This work was supported by grants from the Swedish Research Council (VR), Carl Trygger Foundation (CTS), Magn. Bergvalls Foundation (MBS), and SIDA/SAREC. We thank Dick Sandstro¨m for valuable discussions. References (1) Prestegard, J. H.; Al Hashimi, H. M.; Tolman, J. R. Q. ReV. Biophys. 2000, 33, 371. (2) Thompson, G. S.; Shimizu, H.; Homans, S. W.; Donohue-Rolfe, A. Biochemistry 2000, 39, 13153. (3) Staaf, M.; Ho¨o¨g, C.; Stevensson, B.; Maliniak, A.; Widmalm, G. Biochemistry 2001, 40, 3623. (4) Lycknert, K.; Maliniak, A.; Widmalm, G. J. Phys. Chem. A 2001, 105, 5119. (5) Emsley, J. W.; Luckhurst, G. R.; Stockley, C. P. Proc. R. Soc. London A 1982, 381, 117. (6) Catalano, D.; Di Bari, L.; Veracini, C. A.; Shilstone, G. N.; Zannoni, C. J. Chem. Phys. 1991, 94, 3928. (7) Sandstro¨m, D.; Levitt, M. H. J. Am. Chem. Soc. 1996, 118, 6966. (8) La Penna, G.; Foord, E. K.; Emsley, J. W.; Tildesley, D. J. J. Chem. Phys. 1996, 104, 233. (9) Stevensson, B.; Komolkin, A. V.; Sandstro¨m, D.; Maliniak, A. J. Chem. Phys. 2001, 114, 2332. (10) Tian, F.; Al Hashimi, H. M.; Craighead, J. L.; Prestegard, J. H. J. Am. Chem. Soc. 2001, 123, 485. (11) Cloran, F.; Carmichael, I.; Serianni, A. S. J. Am. Chem. Soc. 1999, 121, 9843. (12) Levitt, M. H. Spin Dynamics: Basics of Nuclear Magnetic Resonance; Wiley: Chichester, 2001. (13) Hardy, B. J.; Egan, W.; Widmalm, G. Int. J. Biol. Macromol. 1995, 17, 149. (14) Tjandra, N.; Bax, A. J. Magn. Reson. 1997, 124, 512. (15) Delaglio, F.; Wu, Z. R.; Bax, A. J. Magn. Reson. 2001, 149, 276. (16) Kjellberg, A.; Widmalm, G. Biopolymers 1999, 50, 391. (17) Nishida, T.; Widmalm, G.; Sandor, P. Magn. Reson. Chem. 1996, 34, 377.
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